一种求解大型Lyapunov矩阵方程的预处理并行算法  

A Preconditioned Parallel Method for Solving Large Lyapunov Matrix Equation

作　　者：侯俊霞[1] 吕全义[1] 曹方颖[1] 谢公南[2] 

机构地区：[1]西北工业大学应用数学系,西安710129 [2]西北工业大学机电学院,工程仿真与宇航计算技术联合实验室,西安710072

出　　处：《应用数学和力学》2013年第5期454-461,共8页Applied Mathematics and Mechanics

基　　金：国家自然科学基金资助项目(11202164);陕西省自然科学基金资助项目(2009JM1008)

摘　　要：研究了一种求解大型Lyapunov矩阵方程的并行预处理变形共轭梯度法.首先将处理小型矩阵方程的Smith预处理方法引入该问题的求解,将原矩阵方程转变为Stein方程,然后采用变形共轭梯度法并行求解预处理后的矩阵方程.其中遇到的难点是需要确定参数μ及求矩阵(A+μI)的逆.基于估计特征值的Gerschgorin圆定理给出了参数μ的估值,再采用变形共轭梯度法并行求得矩阵(A+μI)的逆,从而形成预处理后的矩阵方程.通过数值试验,该算法与未预处理的变形共轭梯度法相比较,预处理算法明显优于未预处理的算法,而且其并行效率高达0.85.A parallel algorithm with preconditioned modified conjugate gradient method for sol- ving large Lyapunov matrix equation was studied. The preconditioned Smith method for small matrix equation was first introduced, and then the modified conjugate gradient method was used for parallel solving the preconditioned Stein matrix equation, which was transformed from the original Lyapunov matrix equation. To fix the involved difficulties such as the determination of the parameterp and the solving inverse matrix of the matrix （A ＋μI）, Gerschgorin theorem and the modified conjugate gradient method were employed. Several numerical experiments show the proposed algorithm is superior to the modified conjugate gradient without precondi-tion. The parallel efficiency is up to 0.85.

关 键 词：LYAPUNOV矩阵方程 并行计算 变形共轭梯度法 预处理方法 

分 类 号：O246[理学—计算数学]